This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A035513 #212 Jun 26 2025 15:27:15
%S A035513 1,2,4,3,7,6,5,11,10,9,8,18,16,15,12,13,29,26,24,20,14,21,47,42,39,32,
%T A035513 23,17,34,76,68,63,52,37,28,19,55,123,110,102,84,60,45,31,22,89,199,
%U A035513 178,165,136,97,73,50,36,25,144,322,288,267,220,157,118,81,58,41,27,233,521
%N A035513 Wythoff array read by falling antidiagonals.
%C A035513 T(0,0)=1, T(0,1)=2,...; y^2-x^2-xy<y if and only if there exist (i,j) with x=T(i,2j) and y=T(i,2j+1). - Claude Lenormand (claude.lenormand(AT)free.fr), Mar 17 2001
%C A035513 Inverse of sequence A064274 considered as a permutation of the nonnegative integers. - _Howard A. Landman_, Sep 25 2001
%C A035513 The Wythoff array W consists of all the Wythoff pairs (x(n),y(n)), where x=A000201 and y=A001950, so that W contains every positive integer exactly once. The differences T(i,2j+1)-T(i,2j) form the Wythoff difference array, A080164, which also contains every positive integer exactly once. - _Clark Kimberling_, Feb 08 2003
%C A035513 For n>2 the determinant of any n X n contiguous subarray of A035513 (as a square array) is 0. - _Gerald McGarvey_, Sep 18 2004
%C A035513 From _Clark Kimberling_, Nov 14 2007: (Start)
%C A035513 Except for initial terms in some cases:
%C A035513 (Row 1) = A000045
%C A035513 (Row 2) = A000032
%C A035513 (Row 3) = A006355
%C A035513 (Row 4) = A022086
%C A035513 (Row 5) = A022087
%C A035513 (Row 6) = A000285
%C A035513 (Row 7) = A022095
%C A035513 (Row 8) = A013655 (sum of Fibonacci and Lucas numbers)
%C A035513 (Row 9) = A022112
%C A035513 (Row 10-19) = A022113, A022120, A022121, A022379, A022130, A022382, A022088, A022136, A022137, A022089
%C A035513 (Row 20-28) = A022388, A022096, A022090, A022389, A022097, A022091, A022390, A022098, A022092
%C A035513 (Column 1) = A003622 = AA Wythoff sequence
%C A035513 (Column 2) = A035336 = BA Wythoff sequence
%C A035513 (Column 3) = A035337 = ABA Wythoff sequence
%C A035513 (Column 4) = A035338 = BBA Wythoff sequence
%C A035513 (Column 5) = A035339 = ABBA Wythoff sequence
%C A035513 (Column 6) = A035340 = BBBA Wythoff sequence
%C A035513 Main diagonal = A020941. (End)
%C A035513 The Wythoff array is the dispersion of the sequence given by floor(n*x+x-1), where x=(golden ratio). See A191426 for a discussion of dispersions. - _Clark Kimberling_, Jun 03 2011
%C A035513 If u and v are finite sets of numbers in a row of the Wythoff array such that (product of all the numbers in u) = (product of all the numbers in v), then u = v. See A160009 (row 1 products), A274286 (row 2), A274287 (row 3), A274288 (row 4). - _Clark Kimberling_, Jun 17 2016
%C A035513 All columns of the Wythoff array are compound Wythoff sequences. This follows from the main theorem in the 1972 paper by Carlitz, Scoville and Hoggatt. For an explicit expression see Theorem 10 in Kimberling's paper from 2008 in JIS. - _Michel Dekking_, Aug 31 2017
%C A035513 The Wythoff array can be viewed as an infinite graph over the set of nonnegative integers, built as follows: start with an empty graph; for all n = 0, 1, ..., create an edge between n and the sum of the degrees of all i < n. Finally, remove vertex 0. In the resulting graph, the connected components are chains and correspond to the rows of the Wythoff array. - _Luc Rousseau_, Sep 28 2017
%C A035513 Suppose that h < k are consecutive terms in a row of the Wythoff array. If k is in an even numbered column, then h = floor(k/tau); otherwise, h = -1 + floor(k/tau). - _Clark Kimberling_, Mar 05 2020
%C A035513 From _Clark Kimberling_, May 26 2020: (Start)
%C A035513 For k > = 0, column k shows the numbers m having F(k+1) as least term in the Zeckendorf representation of m. For n >= 1, let r(n,k) be the number of terms in column k that are <= n. Then n/r(n,k) = n/(F(k+1)*tau + F(k)*(n-1)), by Bottomley's formula, so that the limiting ratio is 1/(F(k+1)*tau + F(k)). Summing over all k gives Sum_{k>=0} 1/(F(k+1)*tau + F(k)) = 1. Thus, in the limiting sense:
%C A035513 38.19...% of the numbers m have least term 1;
%C A035513 23.60...% have least term 2;
%C A035513 14.58...% have least term 3;
%C A035513 9.01...% have least term 5, etc. (End)
%C A035513 Named after the Dutch mathematician Willem Abraham Wythoff (1865-1939). - _Amiram Eldar_, Jun 11 2021
%C A035513 From _Clark Kimberling_, Jun 04 2025: (Start)
%C A035513 Let u(n) = (T(n,1),T(n,2)) mod 2. The positive integers (A000027) are partitioned into 4 sets (sequences):
%C A035513 {n : u(n) = (0,0)} = (3, 5, 9, 15, 19, 25, 29,...) = 1 + 2*A190429
%C A035513 {n: u(n) = (0,1)} = (2, 6, 12, 16, 18, 22, 28,...) = A191331
%C A035513 {n : u(n) = (1,0)} = (1, 7, 11, 13, 17, 21, 23,...) = A086843
%C A035513 {n: u(n) = (1,1)} = (4, 8, 10, 14, 20, 24, 26,...) = A191330.
%C A035513 Let v(n) = (T(n,1),T(n,2)) mod 3. The positive integers are partitioned into 9 sets (sequences):
%C A035513 {n : v(n) = (0,0)} = (4, 13, 19, 28, 43, 52,...) = 1 + 3*A190434
%C A035513 {n: v(n) = (0,1)} = (3, 12, 27, 36, 42, 51,...) = 3*A140399
%C A035513 {n : v(n) = (0,2)} = (5, 11, 20, 35, 44, 50,...) = 2 + 3*A190439
%C A035513 {n: v(n) = (1,0)} = (9, 18, 24, 33, 48, 57,...) = 3*A140400
%C A035513 {n: v(n) = (1,1)} = (2, 8, 17, 26, 32, 41,...) = A384601
%C A035513 {n : v(n) = (1,2)} = (1, 10, 16, 25, 34, 40,...) = A384602
%C A035513 {n: v(n) = (2,0)} = (14, 23, 29, 38, 47, 53,...) = 2 + 3*A190438
%C A035513 {n: v(n) = (2,1)} = (7, 22, 31, 37, 46, 61,...) = 1 + 3*A190433
%C A035513 {n : v(n) = (2,2)} = (6, 15, 21, 30, 39, 45,...) = 3*A140398.
%C A035513 Conjecture: If m >= 2, then {(T(n,1), T(n,2)) mod m} has cardinality m^2. (End)
%D A035513 John H. Conway, Posting to Math Fun Mailing List, Nov 25 1996.
%D A035513 Clark Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
%H A035513 Alois P. Heinz, <a href="/A035513/b035513.txt">Table of n, a(n) for n = 1..5151</a>
%H A035513 Peter G. Anderson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-5/Anderson.pdf">More Properties of the Zeckendorf Array</a>, Fib. Quart. 52-5 (2014), 15-21.
%H A035513 L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-1/carlitz1.pdf">Fibonacci representations</a>, Fib. Quart., Vol. 10, No. 1 (1972), pp. 1-28.
%H A035513 Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/questions/183186/new-order-5-where-fibonacci-and-beatty-meet-at-wythoff">New Order #5: where Fibonacci and Beatty meet at Wythoff</a>.
%H A035513 J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, and N. J. A. Sloane, <a href="/A269725/a269725.txt">On Kimberling Sums and Para-Fibonacci Sequences</a>, Correspondence and Postings to Math-Fun Mailing List, Nov 1996 to Jan 1997.
%H A035513 John Conway and Alex Ryba, <a href="https://doi.org/10.1007/s00283-015-9582-5">The extra Fibonacci series and the Empire State Building</a>, Math. Intelligencer, Vol. 38, No. 1 (2016), pp. 41-48.
%H A035513 Larry Ericksen and Peter G. Anderson, <a href="http://www.cs.rit.edu/~pga/k-zeck.pdf">Patterns in differences between rows in k-Zeckendorf arrays</a>, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18. - _N. J. A. Sloane_, Jun 10 2012
%H A035513 Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions</a>.
%H A035513 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/33-1/kimberling.pdf">The Zeckendorf array equals the Wythoff array</a>, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 3-8.
%H A035513 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Kimberling/kimberling719a.html">Complementary equations and Wythoff Sequences</a>, JIS, Vol. 11 (2008), Article 08.3.3.
%H A035513 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Kimberling/kimber12.html">Lucas Representations of Positive Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
%H A035513 Clark Kimberling and Kenneth B. Stolarsky, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.3.267">Slow Beatty sequences, devious convergence, and partitional divergence</a>, Amer. Math. Monthly, 123, No. 2 (2016), pp. 267-273.
%H A035513 Stéphane Legendre, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/53-2/Legendre10222014.pdf">Labeled Fibonacci Trees</a>, Fibonacci Quart. 53 (2015), no. 2, 152-167.
%H A035513 A. J. Macfarlane, <a href="https://arxiv.org/abs/2405.18128">On the fibbinary numbers and the Wythoff array</a>, arXiv:2405.18128 [math.CO], 2024. See pages 1-2.
%H A035513 Casey Mongoven, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from175to192.pdf">Sonification of multiple Fibonacci-related sequences</a>, Annales Mathematicae et Informaticae, Vol. 41 (2013), pp. 175-192.
%H A035513 N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A035513 N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>
%H A035513 Sam Vandervelde, <a href="http://myslu.stlawu.edu/~svanderv/fibseqnorm.pdf">On the divisibility of Fibonacci sequences by primes of index two</a>, The Fibonacci Quarterly, Vol. 50, No. 3 (2012), pp. 207-216. See Figure 1.
%H A035513 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WythoffArray.html">Wythoff Array</a>.
%H A035513 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wythoff_array">Wythoff array</a>.
%H A035513 Pedro Zanetti, <a href="/A035513/a035513.txt">C++ code snippet, for generating this sequence</a>.
%H A035513 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A035513 T(n, k) = Fib(k+1)*floor[n*tau]+Fib(k)*(n-1) where tau = (sqrt(5)+1)/2 = A001622 and Fib(n) = A000045(n). - _Henry Bottomley_, Dec 10 2001
%F A035513 T(n,-1) = n-1. T(n,0) = floor(n*tau). T(n,k) = T(n,k-1) + T(n,k-2) for k>=1. - _R. J. Mathar_, Sep 03 2016
%e A035513 The Wythoff array begins:
%e A035513 1 2 3 5 8 13 21 34 55 89 144 ...
%e A035513 4 7 11 18 29 47 76 123 199 322 521 ...
%e A035513 6 10 16 26 42 68 110 178 288 466 754 ...
%e A035513 9 15 24 39 63 102 165 267 432 699 1131 ...
%e A035513 12 20 32 52 84 136 220 356 576 932 1508 ...
%e A035513 14 23 37 60 97 157 254 411 665 1076 1741 ...
%e A035513 17 28 45 73 118 191 309 500 809 1309 2118 ...
%e A035513 19 31 50 81 131 212 343 555 898 1453 2351 ...
%e A035513 22 36 58 94 152 246 398 644 1042 1686 2728 ...
%e A035513 25 41 66 107 173 280 453 733 1186 1919 3105 ...
%e A035513 27 44 71 115 186 301 487 788 1275 2063 3338 ...
%e A035513 ...
%e A035513 The extended Wythoff array has two extra columns, giving the row number n and A000201(n), separated from the main array by a vertical bar:
%e A035513 0 1 | 1 2 3 5 8 13 21 34 55 89 144 ...
%e A035513 1 3 | 4 7 11 18 29 47 76 123 199 322 521 ...
%e A035513 2 4 | 6 10 16 26 42 68 110 178 288 466 754 ...
%e A035513 3 6 | 9 15 24 39 63 102 165 267 432 699 1131 ...
%e A035513 4 8 | 12 20 32 52 84 136 220 356 576 932 1508 ...
%e A035513 5 9 | 14 23 37 60 97 157 254 411 665 1076 1741 ...
%e A035513 6 11 | 17 28 45 73 118 191 309 500 809 1309 2118 ...
%e A035513 7 12 | 19 31 50 81 131 212 343 555 898 1453 2351 ...
%e A035513 8 14 | 22 36 58 94 152 246 398 644 1042 1686 2728 ...
%e A035513 9 16 | 25 41 66 107 173 280 453 733 1186 1919 3105 ...
%e A035513 10 17 | 27 44 71 115 186 301 487 788 1275 2063 3338 ...
%e A035513 11 19 | 30 49 79 ...
%e A035513 12 21 | 33 54 87 ...
%e A035513 13 22 | 35 57 92 ...
%e A035513 14 24 | 38 62 ...
%e A035513 15 25 | 40 65 ...
%e A035513 16 27 | 43 70 ...
%e A035513 17 29 | 46 75 ...
%e A035513 18 30 | 48 78 ...
%e A035513 19 32 | 51 83 ...
%e A035513 20 33 | 53 86 ...
%e A035513 21 35 | 56 91 ...
%e A035513 22 37 | 59 96 ...
%e A035513 23 38 | 61 99 ...
%e A035513 24 40 | 64 ...
%e A035513 25 42 | 67 ...
%e A035513 26 43 | 69 ...
%e A035513 27 45 | 72 ...
%e A035513 28 46 | 74 ...
%e A035513 29 48 | 77 ...
%e A035513 30 50 | 80 ...
%e A035513 31 51 | 82 ...
%e A035513 32 53 | 85 ...
%e A035513 33 55 | 88 ...
%e A035513 34 56 | 90 ...
%e A035513 35 58 | 93 ...
%e A035513 36 59 | 95 ...
%e A035513 37 61 | 98 ...
%e A035513 38 63 | ...
%e A035513 ...
%e A035513 Each row of the extended Wythoff array also satisfies the Fibonacci recurrence, and may be extended to the left using this recurrence backwards.
%e A035513 From _Peter Munn_, Jun 11 2021: (Start)
%e A035513 The Wythoff array appears to have the following relationship to the traditional Fibonacci rabbit breeding story, modified for simplicity to be a story of asexual reproduction.
%e A035513 Give each rabbit a number, 0 for the initial rabbit.
%e A035513 When a new round of rabbits is born, allocate consecutive numbers according to 2 rules (the opposite of many cultural rules for inheritance precedence): (1) newly born child of Rabbit 0 gets the next available number; (2) the descendants of a younger child of any given rabbit precede the descendants of an older child of the same rabbit.
%e A035513 Row n of the Wythoff array lists the children of Rabbit n (so Rabbit 0's children have the Fibonacci numbers: 1, 2, 3, 5, ...). The generation tree below shows rabbits 0 to 20. It is modified so that each round of births appears on a row.
%e A035513 0
%e A035513 :
%e A035513 ,-------------------------:
%e A035513 : :
%e A035513 ,---------------: 1
%e A035513 : : :
%e A035513 ,--------: 2 ,---------:
%e A035513 : : : : :
%e A035513 ,-----: 3 ,-----: ,-----: 4
%e A035513 : : : : : : : :
%e A035513 ,--: 5 ,--: ,---: 6 ,---: 7 ,---:
%e A035513 : : : : : : : : : : : : :
%e A035513 ,--: 8 ,--: ,--: 9 ,--: 10 ,--: ,--: 11 ,--: ,--: 12
%e A035513 : : : : : : : : : : : : : : : : : : : : :
%e A035513 : 13 : : 14 : 15 : : 16 : : 17 : 18 : : 19 : 20 :
%e A035513 The extended array's nontrivial extra column (A000201) gives the number that would have been allocated to the first child of Rabbit n, if Rabbit n (and only Rabbit n) had started breeding one round early.
%e A035513 (End)
%p A035513 W:= proc(n,k) Digits:= 100; (Matrix([n, floor((1+sqrt(5))/2* (n+1))]). Matrix([[0,1], [1,1]])^(k+1))[1,2] end: seq(seq(W(n, d-n), n=0..d), d=0..10); # _Alois P. Heinz_, Aug 18 2008
%p A035513 A035513 := proc(r, c)
%p A035513 option remember;
%p A035513 if c = 1 then
%p A035513 A003622(r) ;
%p A035513 else
%p A035513 A022342(1+procname(r, c-1)) ;
%p A035513 end if;
%p A035513 end proc:
%p A035513 seq(seq(A035513(r,d-r),r=1..d-1),d=2..15) ; # _R. J. Mathar_, Jan 25 2015
%t A035513 W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k]; Table[ W[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten
%o A035513 (PARI) T(n,k)=(n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k)
%o A035513 for(k=0,9,for(n=1,k, print1(T(n,k+1-n)", "))) \\ _Charles R Greathouse IV_, Mar 09 2016
%o A035513 (Python)
%o A035513 from sympy import fibonacci as F, sqrt
%o A035513 import math
%o A035513 tau = (sqrt(5) + 1)/2
%o A035513 def T(n, k): return F(k + 1)*int(math.floor(n*tau)) + F(k)*(n - 1)
%o A035513 for n in range(1, 11): print([T(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Apr 23 2017
%o A035513 (Python)
%o A035513 from math import isqrt, comb
%o A035513 from gmpy2 import fib2
%o A035513 def A035513(n):
%o A035513 a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
%o A035513 x = n-comb(a,2)
%o A035513 b, c = fib2(a-x+2)
%o A035513 return b*(x+isqrt(5*x*x)>>1)+c*(x-1) # _Chai Wah Wu_, Jun 26 2025
%Y A035513 See comments above for more cross-references.
%Y A035513 Cf. A003622, A064274 (inverse), A083412 (transpose), A000201, A001950, A080164, A003603, A265650, A019586 (row that contains n).
%Y A035513 For two versions of the extended Wythoff array, see A287869, A287870.
%K A035513 nonn,tabl,easy,nice
%O A035513 1,2
%A A035513 _N. J. A. Sloane_
%E A035513 Comments about the extended Wythoff array added by _N. J. A. Sloane_, Mar 07 2016